We study the structure of the partial order induced by the definability relation on definitions of truth for the language of arithmetic. Formally, a definition of truth is any sentence $$\alpha $$ which extends a weak arithmetical theory (which we take to be $${{\,\mathrm{I\Delta _{0}+\exp }\,}}$$ ) such that for some formula $$\Theta $$ and any arithmetical sentence $$\varphi $$, $$\Theta (\ulcorner \varphi \urcorner )\equiv \varphi $$ is provable in $$\alpha $$. We say that a sentence $$\beta $$ is definable (...) in a sentence $$\alpha $$, if there exists an unrelativized translation from the language of $$\beta $$ to the language of $$\alpha $$ which is identity on the arithmetical symbols and such that the translation of $$\beta $$ is provable in $$\alpha $$. Our main result is that the structure consisting of truth definitions which are conservative over the basic arithmetical theory forms a countable universal distributive lattice. Additionally, we generalize the result of Pakhomov and Visser showing that the set of (Gödel codes of) definitions of truth is not $$\Sigma _2$$ -definable in the standard model of arithmetic. We conclude by remarking that no $$\Sigma _2$$ -sentence, satisfying certain further natural conditions, can be a definition of truth for the language of arithmetic. (shrink)

This paper is an investigation into what could be a goodexplication of ``theory S is reducible to theory T''''. Ipresent an axiomatic approach to reducibility, which is developedmetamathematically and used to evaluate most of the definitionsof ``reducible'''' found in the relevant literature. Among these,relative interpretability turns out to be most convincing as ageneral reducibility concept, proof-theoreticalreducibility being its only serious competitor left. Thisrelation is analyzed in some detail, both from the point of viewof the reducibility axioms and of modal logic.